Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2025 • Question 5

Question

Math 6420 • PDE Fall 2025 • Question 5

Let \(f:\mathbb {R}^d\to \mathbb {R}\) smooth. Show that there is at most one smooth solution of \[ \begin {cases} -\Delta u=f & \text {in } \mathbb {R}^d,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0. \end {cases} \]

Give a (non-trivializing) additional condition on \(f\) so that you can show existence of such a solution.

Raw LaTeX

Let \(f:\mathbb{R}^d\to\mathbb{R}\) smooth. Show that there is at most one
smooth solution of
\[
\begin{cases}
-\Delta u=f & \text{in } \mathbb{R}^d,\\
\displaystyle\lim_{|x|\to\infty}u(x)=0.
\end{cases}
\]

Give a (non-trivializing) additional condition on \(f\) so that you can show
existence of such a solution.

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Fundamental Solutions